QJEEAdv22Ma8b25: JEE Advanced 2022 Mathematics - Class 11 JEEAdv22Ma8b25 Solution: Consider the hyperbola with foci at and , where lies on the…

From property we know, tangent and normal is bisector of the angle between focal radii.

$\therefore$ Tangent AB divides the angle $\angle SP{S_1} = \alpha$ equal parts.

From another property, we know, if we draw perpendicular to the tangent on the hyperbola from two foci, then product of length of the perpendicular from foci = b²

$\therefore$ $l \times \delta = {b^2}$

Given hyperbola, ${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$

$\therefore$ ${a^2} = 100$

and ${b^2} = 64$

$\therefore$ $l \times \delta = 64$ ....... (1)

From right angle triangle S₁ MP we get, $\sin {\alpha \over 2} = {l \over \beta }$

$\therefore$ $l = \beta \sin {\alpha \over 2}$ ....... (2)

Putting value of l in equation (1), we get

$\left( {\beta \sin {\alpha \over 2}} \right) \times \delta = 64$

$\Rightarrow \delta \beta \sin {\alpha \over 2} = 64$

$\therefore$ ${{\beta \delta } \over 9}\sin {\alpha \over 2}$

$= {{64} \over 9} = 7.1$

$\therefore$ Greatest integer $= [7.1] = 7$